Ebook: Protoalgebraic Logics

The main aim of this book is to present recent ideas in logic centered around the notion of a consequence operation. We wish to show these ideas in a factually and materially connected way, i.e., in the form of a consistent theory derived from several simple assumptions and definitions. These ideas have arisen in many research centers. The thorough study of their history can certainly be an exciting task for the historian of logic; in the book this aspect of the theory is being played down. The book belongs to abstract algebraic logic, the area of research that explores to a large extent interconnections between algebra and logic. The results presented here concern logics defined in zero-order languages (Le., quantifier-free sentential languages without predicate symbols). The reach of the theory expounded in the book is, in fact, much wider. The theory is also valid for logics defined in languages of higer orders. The problem of transferring the theory to the level of first-order languages has been satisfactorily solved and new ideas within this area have been put forward in the work of Blok and Pigozzi [1989].

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The main aim of this monograph is to provide a structured study of the algebraic method in metalogic. In contrast to traditional algebraic logic, where the focus is on the algebraic forms of specific deductive systems, abstract algebraic logic is concerned with the process of algebraization itself. This book presents in a systematic way recent ideas in abstract algebraic logic centered around the notion of the Leibniz operator. The stress is put on the taxonomy of deductive systems. Isolating a list of plausible properties of the Leibniz operator serves as a basis for distinguishing certain natural classes of sentential logics. The hierarchy of deductive systems presented in the book comprises, among others, the following classes: protoalgebraic logics, equivalential logics, algebraizable logics, and Fregean logics. Because of the intimate connection between algebraic and logical structures, the book also provides a uniform treatment of various topics concerning deduction theorems and quasivarieties of algebras.
The presentation of the above classes of logics is accompanied by a wealth of examples illustrating the general theory. An essential part of the book is formed by the numerous exercises integrated into the text.
This book is both suitable for logically and algebraically minded graduate and advanced graduate students of mathematics, computer science and philosophy, and as a reference work for the expert.

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The main aim of this monograph is to provide a structured study of the algebraic method in metalogic. In contrast to traditional algebraic logic, where the focus is on the algebraic forms of specific deductive systems, abstract algebraic logic is concerned with the process of algebraization itself. This book presents in a systematic way recent ideas in abstract algebraic logic centered around the notion of the Leibniz operator. The stress is put on the taxonomy of deductive systems. Isolating a list of plausible properties of the Leibniz operator serves as a basis for distinguishing certain natural classes of sentential logics. The hierarchy of deductive systems presented in the book comprises, among others, the following classes: protoalgebraic logics, equivalential logics, algebraizable logics, and Fregean logics. Because of the intimate connection between algebraic and logical structures, the book also provides a uniform treatment of various topics concerning deduction theorems and quasivarieties of algebras.
The presentation of the above classes of logics is accompanied by a wealth of examples illustrating the general theory. An essential part of the book is formed by the numerous exercises integrated into the text.
This book is both suitable for logically and algebraically minded graduate and advanced graduate students of mathematics, computer science and philosophy, and as a reference work for the expert.
Content:
Front Matter....Pages i-xi
Introduction....Pages 1-20
Basic Definitions and Facts....Pages 21-66
Front Matter....Pages 67-67
Protoalgebraic Logics....Pages 69-122
Protoalgebraicity and the Deduction Theorem....Pages 123-184
Equivalential Logics....Pages 185-242
Front Matter....Pages 243-243
Quasivarieties of Algebras....Pages 245-285
Algebraizable Logics....Pages 287-345
Regularly Algebraizable Logics....Pages 347-371
Fregean Logics....Pages 373-418
Back Matter....Pages 419-452

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The main aim of this monograph is to provide a structured study of the algebraic method in metalogic. In contrast to traditional algebraic logic, where the focus is on the algebraic forms of specific deductive systems, abstract algebraic logic is concerned with the process of algebraization itself. This book presents in a systematic way recent ideas in abstract algebraic logic centered around the notion of the Leibniz operator. The stress is put on the taxonomy of deductive systems. Isolating a list of plausible properties of the Leibniz operator serves as a basis for distinguishing certain natural classes of sentential logics. The hierarchy of deductive systems presented in the book comprises, among others, the following classes: protoalgebraic logics, equivalential logics, algebraizable logics, and Fregean logics. Because of the intimate connection between algebraic and logical structures, the book also provides a uniform treatment of various topics concerning deduction theorems and quasivarieties of algebras.
The presentation of the above classes of logics is accompanied by a wealth of examples illustrating the general theory. An essential part of the book is formed by the numerous exercises integrated into the text.
This book is both suitable for logically and algebraically minded graduate and advanced graduate students of mathematics, computer science and philosophy, and as a reference work for the expert.
Content:
Front Matter....Pages i-xi
Introduction....Pages 1-20
Basic Definitions and Facts....Pages 21-66
Front Matter....Pages 67-67
Protoalgebraic Logics....Pages 69-122
Protoalgebraicity and the Deduction Theorem....Pages 123-184
Equivalential Logics....Pages 185-242
Front Matter....Pages 243-243
Quasivarieties of Algebras....Pages 245-285
Algebraizable Logics....Pages 287-345
Regularly Algebraizable Logics....Pages 347-371
Fregean Logics....Pages 373-418
Back Matter....Pages 419-452
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